Faster algorithms for approximate distance oracles and all-pairs small stretch paths

`Let G = (V, E) be a weighted undirected graph with vertical bar V vertical bar = n and vertical bar E vertical bar = m. An estimate delta(u, v) of the distance delta (u, v) in G between u, v is an element of V is said to be of stretch t iff delta(u, v) <= delta(u, v) <= t (.) b(u, v). The most efficient algorithms known for computing small stretch distances in G are the approximate distance oracles of [16] and the three algorithms in [9] to compute all-pairs stretch t distances for t = 2; 7/3, and 3. We present faster algorithms for these problems. For any integer k > 1, Thorup and Zwick in [16] gave an O(kmn(1/k)) algorithm to construct a data structure of size O(kn(1+1/k)) which, given a query (u, v) is an element of V x V, returns in O(k) time, a 2k - 1 stretch estimate of delta(u, v). But for small values of k, the time to construct the oracle is rather high. Here we present an O(n(2) log n) algorithm to construct such a data structure of size O(kn(1+1/k))for all integers k >= 2. Our query answering time is O(k) for k > 2 and 0 (log n) for k = 2. We use a new generic scheme for all-pairs approximate shortest paths for these results. This scheme also enables us to design faster algorithms for all-pairs t-stretch distances for t = 2 and 7/3, and compute all-pairs almost stretch 2 distances in O(n(2) log n) time.